Abstract
Recent interest in Nash equilibria led to a study of the price of anarchy (poa) and the strong price of anarchy (spoa) for scheduling problems. The two measures express the worst case ratio between the cost of an equilibrium (a pure Nash equilibrium, and a strong equilibrium, respectively) to the cost of a social optimum.
We consider scheduling on uniformly related machines. Here the atomic players are the jobs, and the delay of a job is the completion time of the machine running it, also called the load of this machine. The social goal is to minimize the maximum delay of any job, while the selfish goal of each job is to minimize its own delay, that is, the delay of the machine running it.
While previous studies either consider identical speed machines or an arbitrary number of speeds, focusing on the number of machines as a parameter, we consider the situation in which the number of different speeds is small. We reveal a linear dependence between the number of speeds and the poa. For a set of machines of at most p speeds, the poa turns out to be exactly p + 1. The growth of the poa for large numbers of related machines is therefore a direct result of the large number of potential speeds. We further consider a well known structure of processors, where all machines are of the same speed except for one possibly faster machine. We investigate the poa as a function of both the speed of the fastest machine and the number of slow machines, and give tight bounds for nearly all cases.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Andelman, N., Feldman, M., Mansour, Y.: Strong price of anarchy. In: Proc. of the 18th ACM-SIAM Symposium on Discrete Algorithms (SODA 2007), pp. 189–198 (2007)
Cho, Y., Sahni, S.: Bounds for List Schedules on Uniform Processors. SIAM Journal on Computing 9(1), 91–103 (1980)
Czumaj, A.: Selfish routing on the internet. In: Leung, J. (ed.) Handbook of Scheduling: Algorithms, Models, and Performance Analysis, vol. 42, CRC Press, Boca Raton (2004)
Czumaj, A., Vöcking, B.: Tight bounds for worst-case equilibria. ACM Transactions on Algorithms 3(1) (2007)
Epstein, L.: Equilibria for two parallel links: The strong price of anarchy versus the price of anarchy. manuscript (2007)
Even-Dar, E., Kesselman, A., Mansour, Y.: Convergence time to nash equilibria. In: Baeten, J.C.M., Lenstra, J.K., Parrow, J., Woeginger, G.J. (eds.) ICALP 2003. LNCS, vol. 2719, pp. 502–513. Springer, Heidelberg (2003)
Feldmann, R., Gairing, M., Lücking, T., Monien, B., Rode, M.: Nashification and the coordination ratio for a selfish routing game. In: Baeten, J.C.M., Lenstra, J.K., Parrow, J., Woeginger, G.J. (eds.) ICALP 2003. LNCS, vol. 2719, pp. 514–526. Springer, Heidelberg (2003)
Fiat, A., Kaplan, H., Levy, M., Olonetsky, S.: Strong price of anarchy for machine load balancing. In: Arge, L., Cachin, C., Jurdziński, T., Tarlecki, A. (eds.) ICALP 2007. LNCS, vol. 4596, pp. 583–594. Springer, Heidelberg (2007)
Finn, G., Horowitz, E.: A linear time approximation algorithm for multiprocessor scheduling. BIT Numerical Mathematics 19(3), 312–320 (1979)
Fotakis, D., Kontogiannis, S.C., Koutsoupias, E., Mavronicolas, M., Spirakis, P.G.: The structure and complexity of nash equilibria for a selfish routing game. In: Widmayer, P., Triguero, F., Morales, R., Hennessy, M., Eidenbenz, S., Conejo, R. (eds.) ICALP 2002. LNCS, vol. 2380, pp. 124–134. Springer, Heidelberg (2002)
Gonzalez, T., Ibarra, O.H., Sahni, S.: Bounds for LPT Schedules on Uniform Processors. SIAM Journal on Computing 6(1), 155–166 (1977)
Holzman, R., Law-Yone, N.: Strong equilibrium in congestion games. Games and Economic Behavior 21(1-2), 85–101 (1997)
Koutsoupias, E., Mavronicolas, M., Spirakis, P.G.: Approximate equilibria and ball fusion. Theory of Computing Systems 36(6), 683–693 (2003)
Koutsoupias, E., Papadimitriou, C.H.: Worst-Case Equilibria. In: Meinel, C., Tison, S. (eds.) STACS 1999. LNCS, vol. 1563, Springer, Heidelberg (1999)
Kovács, A.: Tighter Approximation Bounds for LPT Scheduling in Two Special Cases. In: Calamoneri, T., Finocchi, I., Italiano, G.F. (eds.) CIAC 2006. LNCS, vol. 3998, pp. 187–198. Springer, Heidelberg (2006)
Li, R., Shi, L.: An on-line algorithm for some uniform processor scheduling. SIAM Journal on Computing 27(2), 414–422 (1998)
Liu, J.W.S., Liu, C.L.: Bounds on scheduling algorithms for heterogeneous computing systems. In: Becvar, J. (ed.) MFCS 1979. LNCS, vol. 74, pp. 349–353. Springer, Heidelberg (1979)
Mavronicolas, M., Spirakis, P.G.: The price of selfish routing. In: Proc. of the 33rd Annual ACM Symposium on Theory of Computing (STOC 2001), pp. 510–519 (2001)
Nisan, N., Ronen, A.: Algorithmic mechanism design. Games and Economic Behavior 35, 166–196 (2001)
Papadimitriou, C.H.: Algorithms, games, and the internet. In: Proc. of the 33rd Annual ACM Symposium on Theory of Computing (STOC 2001), pp. 749–753 (2001)
Roughgarden, T.: Selfish routing and the price of anarchy. MIT Press, Cambridge (2005)
Roughgarden, T., Tardos, É.: How bad is selfish routing? Journal of the ACM 49(2), 236–259 (2002)
Schuurman, P., Vredeveld, T.: Performance guarantees of local search for multiprocessor scheduling. Informs Journal on Computing 19(1), 52–63 (2007)
Tennenholtz, M., Rozenfeld, O.: Strong and Correlated Strong Equilibria in Monotone Congestion Games. In: Spirakis, P.G., Mavronicolas, M., Kontogiannis, S.C. (eds.) WINE 2006. LNCS, vol. 4286, pp. 74–86. Springer, Heidelberg (2006)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2008 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Epstein, L., van Stee, R. (2008). The Price of Anarchy on Uniformly Related Machines Revisited. In: Monien, B., Schroeder, UP. (eds) Algorithmic Game Theory. SAGT 2008. Lecture Notes in Computer Science, vol 4997. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-79309-0_6
Download citation
DOI: https://doi.org/10.1007/978-3-540-79309-0_6
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-79308-3
Online ISBN: 978-3-540-79309-0
eBook Packages: Computer ScienceComputer Science (R0)